Chapter P: Preliminaries - PDF Free Download

Chapter P: Preliminaries Spring 2018 Department of Mathematics Hong Kong Baptist University 1 / 67

Preliminaries The preliminary chapter reviews the most important things that you should know before beginning calculus. Depending on your pre-calculus background, you may or may not be familiar with the topics in the preliminaries chapter. (1) If you are, you may want to skim over these materials to refresh your understanding of the terms used; (2) If not, you should study this chapter in detail. 2 / 67

P.1 Real Numbers and the Real Line Real numbers are numbers that can be expressed as decimals, for example, 5 = 5.00000... 4 3 = 1.33333... 2 = 1.41421... π = 3.14159... e = 2.71828... The real numbers can be represented geometrically as points on the real line, denoted by R. 3 / 67

Intervals Let a and b be two real numbers and a < b. Then (i) (a, b): the open interval from a to b, consisting of all real numbers x satisfying a < x < b. (ii) [a, b]: the closed interval from a to b, consisting of all real numbers x satisfying a x b. (iii) [a, b): the half-open interval from a to b, consisting of all real numbers x satisfying a x < b. (iv) (a, b]: the half-open interval from a to b, consisting of all real numbers x satisfying a < x b. Remark: Note that the whole real line is also an interval, denoted by R = (, ). The symbol is called infinity. 4 / 67

Unions, Intersections A real number is in the union of intervals if it is in at least one of them. A real number is in the intersection of intervals if it is in all of them. Examples: (a) The number 5 belongs to (, 1) (1, 2) (3, 6]. (b) The union (, 2) (2, ) contains all numbers except 2. (c) The intersection ( 5, 5) [3, ) is equal to the interval [3, 5). 5 / 67

Example 1: Solve (a) x 2 5x + 6 < 0, (b) x 2 + 5x + 6 > 0. Solution: (a) Since x 2 5x + 6 = (x 2)(x 3), we see that the product is negative when exactly one of the factors is negative. Thus, any solution must satisfy { { x 2 > 0 x 2 < 0 OR x 3 < 0 x 3 > 0 2 < x < 3 NO SOLUTION Hence, the solution set is the interval (2, 3). 6 / 67

Example 1: Solve (a) x 2 5x + 6 < 0, (b) x 2 + 5x + 6 > 0. Solution: (b) Since x 2 + 5x + 6 = (x + 2)(x + 3), we see that the product is positive when both factors have the same sign. Thus, any solution must satisfy { { x + 2 > 0 x + 2 < 0 OR x + 3 > 0 x + 3 < 0 x > 2 x < 3 Hence, the solution set is the union of intervals (, 3) ( 2, ). 7 / 67

Absolute Value The absolute value, or magnitude, of a number x, denoted by x, is defined by the formula { x, if x 0 x = x, if x < 0. Properties of absolute values include: 1. a = a. 2. ab = a b and a/b = a / b. 3. a ± b a + b. 4. If D is a positive number, then x a < D a D < x < a + D. x a > D either x < a D or x > a + D. 8 / 67

Example 2: Solve 1 2 x < 3. Solution: We must have 3 < 1 2 x < 3 4 < 2 x < 2 4 > 2 x > 2. We now have two cases: (a) If x > 0, then the two inequalities become { 2x < 2 4x > 2 = { x > 1 x > 1 2 = x > 1 2. (b) If x < 0, we get { 2x > 2 4x < 2 = { x < 1 x < 1 2 = x < 1. Thus, the solution set is (, 1) ( 1 2, ). 9 / 67

P.2 Cartesian Coordinates in the Plane The positions of all points in a plane can be measured with respect to two perpendicular real lines (referred to as x-axis and y-axis) in the plane intersecting at the 0-point of each. The point of intersection of the x-axis and the y-axis is called the origin and is often denoted by the letter O. For a point P(a, b), we call a the x-coordinate of P, and b the y-coordinate of P. The ordered pair (a, b) is called the Cartesian coordinates of the point P. The coordinate axes divide the plane into four regions called quadrants, numbered from I to IV. 10 / 67

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(a) x 2 + y 2 = 4, and (b) x 2 + y 2 4 15 / 67

Example 3: Solve the inequality x 2 5x 6 with the aid of a graph. Solution: We plot the graphs of y = x 2 5x and y = 6, which intersect at ( 1, 6) and (6, 6). Since the curve lies above the horizontal line to the left of x = 1 and to the right of x = 6, we conclude that the solution set is (, 1] [6, ). ( 1, 6) y y = x 2 5x y = 6 (6, 6) x 16 / 67

Straight Lines Given two points P 1 (x 1, y 1 ) and P 2 (x 2, y 2 ) in the plane, there is a unique straight line passing through them both. We call the line L = P 1 P 2. For any nonvertical line L, we define the slope of the line is m = y x = y 2 y 1 x 2 x 1. 17 / 67

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Equations of Straight Lines The point-slope equation of a non-vertical line that passes through the point (x 1, y 1 ) and has slope m is y = m(x x 1 ) + y 1. The horizontal line passing through the point (x 1, y 1 ) is y = y 1. The vertical line passing through the point (x 1, y 1 ) is x = x 1. If an oblique line (i.e., neither horizontal or vertical) cuts the x and y axes at (a, 0) and (0, b) respectively, then a and b are called its x- and y-intercepts. 19 / 67

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Example 4: Find the equation of the line passing through (3, 1) and (1, 2). What is its slope, and what are x- and y-intercepts? Solution: The slope is m = 2 ( 1) 1 3 Thus, the equation is given by = 3 2. y = 3 2 (x 1) + 2 or y = 3 2 x + 7 2. We calculate its x-intercept a by substituting y = 0: 0 = 3 2 a + 7 2 = a = 7 3. The y-intercept b is obtained by substituting x = 0: b = 3 2 0 + 7 2 = b = 7 2. 21 / 67

P.4 Functions and Their Graphs The term function was first used by Leibniz in 1673 to denote the dependence of one quantity on another. For instance, the area A of a circle depends on the radius r according to the formula A = πr 2. We can say that the area (as the dependent variable) is a function of the radius (as the independent variable). As a generic function, we often refer to y as the dependent variable and x as the independent variable. Then to denote that y is a function of x, we write y = f (x). 22 / 67

Definition A function f on a set D into a set S is a rule that assigns a unique element f (x) in S to each element x in D. 23 / 67

Graphs of Functions For any given pair of (x, f (x)), we can find a unique point in the x-y plane to represent it. That is, the function y = f (x) can be represented by a curve in Cartesian coordinates system. y y = f (x) (x, f (x)) x 24 / 67

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Domain and Range Definition (1) The domain of the function f, D(f ), is the set of all possible input values for the independent variable. (2) The range of the function f, R(f ), is the set of all possible output values for the dependent variable. If x is an input for which f (x) has a well-defined answer, then x is in the domain of f. Otherwise, it is not in D(f ). Domain convention: if f is defined by a formula without the domain being specified, then D(f ) is assumed to be the largest set for which the formula makes sense. If y is a value for which the equation y = f (x) has at least one solution x, then y is in the range of f. Otherwise, it is not in R(f ). 27 / 67

Example 5: Find the domain and range of function: y = f (x) = 2x 2 + 1, where x [1, 3]. Solution: (1) The domain of f is given as D(f ) = [1, 3]. (2) The range of f is R(f ) = [3, 19]. 28 / 67

Example 6: Find the domain and the range of function: y = f (x) = 9 x 2 + 1. Solution: (1) The domain of f is D(f ) = [ 3, 3]. (2) The range of f is R(f ) = [1, 4]. 29 / 67

Example 7: Find the range of function f (x) = Solution: x x 2, where x R. + 1 We need to find all y R for which the equation y = x x 2 + 1, or equivalently, yx 2 x + y = 0, has at least one solution x. Now since x is a real number, we have = 1 4y 2 0, or 1 2 y 1 2. We thus conclude that the range of f is R(f ) = [ 0.5, 0.5]. 30 / 67

Example 8: Find the range of function f (x) = Solution: x x 4, where x R. + 1 To find the range of y, we need to learn Calculus. More specifically, the differentiation in Chapter 2 will help us to solve this problem. Before starting differentiation, however, we need to first introduce the limits and continuity of a function in Chapter 1. 31 / 67

Even and Odd Functions Definition Suppose that x belongs to the domain of f whenever x does. (1) We say that f is an even function if f ( x) = f (x) for every x in the domain of f. (2) We say that f is an odd function if f ( x) = f (x) for every x in the domain of f. Remark: (1) The graph of an even function is symmetric about the y-axis. (2) The graph of an odd function is symmetric about the origin. 32 / 67

P.5 Combining Functions to Make New Functions Functions can be combined in a variety of ways to produce new functions. Like numbers, functions can be added, subtracted, multiplied, and divided (when the denominator is nonzero) to produce new functions. There is another method, called composition, by which two functions can be combined to form a new function. We can also define new functions by using different formulas on different parts of its domain, referred to as piecewise defined functions. 33 / 67

Sums, Differences, Products, Quotients, and Multiples Definition Let f and g be two functions and c is a real number. Then for every x that belongs to the domains of both f and g, we define functions f + g, f g, f g, f /g, and cf by the formulas: (f + g)(x) = f (x) + g(x), (f g)(x) = f (x) g(x), (fg)(x) = f (x)g(x), ) ( f g (x) = f (x), where g(x) 0, g(x) (cf )(x) = c f (x). 34 / 67

Example 9: The functions f and g are defined by the formulas f (x) = x and g(x) = 2 x. Find the new functions f + g, f g, fg, f /g, g/f and f 4 at x. Also specify the domain of each function. Solution: (1) (f + g)(x) = x + 2 x, D(f + g) = [0, 2]. (2) (f g)(x) = x 2 x, D(f g) = [0, 2]. (3) (fg)(x) = x(2 x), D(fg) = [0, 2]. (4) (f /g)(x) = x/(2 x), D(f /g) = [0, 2). (5) (g/f )(x) = (2 x)/x, D(g/f ) = (0, 2]. (6) (f 4 )(x) = x 2, D(f 4 ) = [0, ). 36 / 67

Composite Functions Definition Let f and g be two functions. The composite function of f g is defined by f g(x) = f (g(x)). The domain of f g consists of those numbers x in the domain of g for which g(x) is the domain of f. Remark: (1) If the range of g is contained in the domain of f, then the domain of f g is just the domain of g. (2) g is called the inner function and f the outer function. To calculate f (g(x)), we first calculate g and then calculate f. (3)The functions f g and g f are usually quite different. 37 / 67

The output of g becomes the input of f. 38 / 67

Example 10: Let f (x) = x 2 + 1 and g(x) = x 2. (a) Find the composite function f g and identify its domain; (a) Find the composite function g f and identify its domain. Solution: 39 / 67

Example 10: Let f (x) = x 2 + 1 and g(x) = x 2. (a) Find the composite function f g and identify its domain; (a) Find the composite function g f and identify its domain. Solution: (a) For the first composition, we have (f g)(x) = f ( x 2) = ( x 2) 2 + 1 = x 1. The domain of f g is D(f g) = [2, ). 40 / 67

Piecewise Defined Functions (1) One example is the absolute value function: { x, if x 0, x = x, if x < 0. (2) Another example is the sign function: sgn(x) = x x = 1, if x > 0, 1, if x < 0, undefined, if x = 0. 42 / 67

Piecewise Defined Functions (3) A third example is the function { x, if x 2, g(x) = 1, if x = 2. Remark: A piecewise defined function with two formulas is NOT two separate functions. It is a single function in which different formulas are used to give the answer, depending on the input. 43 / 67

Piecewise Defined Functions (4) The greatest integer function (or the floor function) is a function whose value at x, denoted by x, is defined to be the largest integer that is x. Examples: 2.4 = 2, 1.9 = 1, 0 = 0, 1.2 = 2. 44 / 67

Basic Elementary Functions The following functions are commonly used in Calculus and are the so-called basic elementary functions: 1) Power functions: x a (a is a real number) 2) Exponential functions: a x (a is a positive real number, one important value of a is e = 2.71828) 3) Logarithmic functions: log a (x) (a is a positive real number, with log e (x) denoted by ln(x)) 4) Trigonometric functions: sin(x), cos(x), tan(x), cot(x), sec(x), csc(x) 5) Inverse Trigonometric functions: arcsin(x), arccos(x), arctan(x), arccot(x), arcsec(x), arccsc(x) 45 / 67

Elementary Functions Definition An elementary function is a function of one variable built from a finite number of basic elementary functions and constants through composition and combinations using the four elementary operations (+,,, /), where (f + g)(x) = f (x) + g(x), (f g)(x) = f (x) g(x), (kf )(x) = k f (x) where k R, (f g)(x) = f (x) g(x), (f /g)(x) = f (x)/g(x) where g(x) 0. 46 / 67

Examples of elementary functions include: f (x) = ln(sin(2x 2 + 1)) + e x + 1 The polynomial: P(x) = a n x n + a n 1 x n 1 + + a 1 x + a 0 The rational function: P(x)/Q(x) The absolute value function: f (x) = x 47 / 67

Not all functions are elementary. Examples of non-elementary functions include: The Dirichlet function: { 0 if x is a rational number, f (x) = 1 if x is an irrational number. The error function: f (x) = 2 π x 0 e t2 dt. 48 / 67

Example 11: Show that g(x) = sin( 2 x + 1) is an elementary function. Solution: Consider the basic elementary functions f 1 (x) = 2 x, f 2 (x) = 1, f 3 (u) = u, f 4 (v) = sin(v). Then g can be written as g(x) = f 4 (f 3 (f 1 (x) + f 2 (x))) = f 4 f 3 (f 1 + f 2 )(x). Thus, g is an elementary function. 49 / 67

P.6 Polynomials and Rational Functions Definition A polynomial is a function P whose value at x is P(x) = a n x n + a n 1 x n 1 + + a 1 x + a 0, where a n, a n 1,..., a 1 and a 0, called the coefficients of the polynomial, are constants and, if n > 0, then a n 0. Remark: The number n, the degree of the highest power of x in the polynomial, is called the degree of the polynomial. 50 / 67

Definition If P(x) and Q(x) are two polynomials and Q(x) is not the zero polynomial (i.e., Q(x) 0), then the function R(x) = P(x) Q(x) is called a rational function. A rational function is called proper if the degree of P is strictly lower than the degree of Q. Remarks: By the domain convention, the domain of R(x) consists of all real numbers x except those for which Q(x) = 0. Using the division algorithm, one can write any rational function as the sum of a polynomial and a proper rational function. 51 / 67

Example 12: Write 2x 3 3x 2 + 3x + 4 x 2 + 1 proper rational function. as the sum of a polynomial and a Solution: Long division gives 2x 3 x 2 + 1 2x 3 3x 2 +3x +4 2x 3 +2x 3x 2 + x +4 3x 2 3 x +7 Thus, 2x 3 3x 2 + 3x + 4 x 2 + 1 = 2x 3 + x + 7 x 2 + 1. 52 / 67

The Factor Theorem Theorem Let P be a polynomial of degree at least 1. Then the real number r is a root of P (i.e., P(r) = 0) if and only if (x r) is a factor of P(x). Proof: By the division algorithm, we have P(x) = (x r)q(x) + c, where c is a polynomial of degree 0, i.e., a constant. So P(r) = 0 c = 0 P(x) is divisible by x r. 53 / 67

Example 13: Let P(x) = x 3 3x 2 16x 12. Verify that P( 1) = 0, and express P(x) as a product of linear factors. Solution: We have P( 1) = ( 1) 3 3( 1) 2 16( 1) 12 = 1 3 + 16 12 = 0. Thus, (x + 1) is a factor of P(x), so long division gives x 3 3x 2 16x 12 = (x + 1)(x 2 4x 12) = (x + 1)(x + 2)(x 6). Remark: The relationship between roots and linear factors will be useful when evaluating limits of rational functions in Chapter 1. 54 / 67

The Quadratic Formula Theorem The two solutions of the quadratic equation Ax 2 + Bx + C = 0 where A, B, and C are constants and A 0, are given by x = B ± B 2 4AC 2A. Remark: The quantity = B 2 4AC is called the discriminant of the quadratic equation or polynomial. 55 / 67

P.7 The Trigonometric Functions 56 / 67

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Let C be the circle with center at the origin O and radius 1. Recall that its equation is x 2 + y 2 = 1. We use the arc length t as a measure of the size of the angle AOP t. Specifically, the radian measure of angle AOP t is AOP t = t radians. Note that the angle can also be measured in degrees. Since P π is the point ( 1, 0), halfway around the circle C from the point A, we have AOP π = π and also AOP π = 180 o. Hence, π = 180 o. Some frequently used degrees are, e.g., π/2 = 90 o, π/3 = 60 o, π/4 = 45 o, and π/6 = 30 o. 58 / 67

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Some Useful Identities (1) The range of Sine and Cosine: 1 sin(t) 1 and 1 cos(t) 1. (2) The Pythagorean identity: sin 2 (t) + cos 2 (t) = 1. (3) Periodicity: sin(t + 2π) = sin(t) and cos(t + 2π) = cos(t). 61 / 67

(4) Sine is an odd function, and Cosine is an even function: sin( t) = sin(t) and cos( t) = cos(t). (5) Complementary angle identities: sin( π 2 t) = cos(t) and cos(π 2 t) = sin(t). (6) Supplementary angle identities: sin(π t) = sin(t) and cos(π t) = cos(t). 62 / 67

The graph of sin(x) 63 / 67

The graph of cos(x) 64 / 67

Additional Formulas for Sine and Cosine sin(s + t) = sin(s) cos(t) + cos(s) sin(t), sin(s t) = sin(s) cos(t) cos(s) sin(t), cos(s + t) = cos(s) cos(t) sin(s) sin(t), cos(s t) = cos(s) cos(t) + sin(s) sin(t). 65 / 67

Proof: The last formula, cos(s t) = cos(s) cos(t) + sin(s) sin(t), can be shown by noting that P s P t = P s t A. Then by this formula, we can readily derive the remaining three formulas. [The proof is not required.] 66 / 67

Letting s = t, by the first and third additional formulas we have Further, we can show that sin(2t) = 2 sin(t) cos(t), cos(2t) = cos 2 (t) sin 2 (t). sin 2 (t) = 1 cos(2t), 2 cos 2 (t) = 1 + cos(2t). 2 67 / 67